Concentric sphere diffusion

A melt inclusion and its olivine host are concentric spheres of radius R- and / 2. Consider H2O in the melt inclusion and its re-equilibration with ambient melt outside olivine. Ignore anisotropy of olivine. If / i=0.1mm, / 2 = 1 mm, H2O diffusivity in olivine is D = 10 " m /s, K=0.000, find the reequilibration timescale. 

Solid sphere For isotropic diffusion in a spherical mineral of radius a with uniform initial concentration Cq and zero surface concentration, Ar diffusion profile is as follows (Equation 3-68g) ... 

Diffusion from spherical silicate samples can be studied readily by observing the loss of volatile components of the silicate as a function of time. Where the sphere is initially uniform in composition and subsequent vaporization allows one to assume a zero surface concentration of the vaporizing component thereafter, the solution to the differential equations and boundary conditions governing concentration-independent diffusion is given by... 

Exercise. A particle obeys the ordinary diffusion equation in the space between two concentric spheres. Find the splitting probability and the conditional mean first-passage times. 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

Another model for understanding the diffusion of lipopolymers at the air-water interface in Region II is the free area model, useful for describing the motion of phospholipids on a Langmuir monolayer and many systems where the diffusing particles can be approximated by hard spheres, disks or cylinders [38], In this model, a particle can diffuse in any direction that is free, or in other words, in any direction that is empty of another particle. As would be expected, more crowded or concentrated systems diffuse more slowly. Assuming the particles are at a constant temperature and that other energetic considerations can be described within a constant, D0, this type of diffusion can be expressed as... 

The plateau currents of steady-state voltammograms can also provide the critical dimension of the electrode (e.g., tq for a sphere or disk). When a new UME is constructed, its critical dimension is often not known however, it can be easily determined from a single voltammogram recorded for a solution of a species with a known concentration and diffusion coefficient, such as Ru(NH3)5 [D = 5.3 X 10 cm /s in 0.09 M phosphate buffer, pH 7.4 (8)]. 

A halfway bounce-back boundary condition was imposed at the spheres/fluid interface. Gas particles (molecules) which are propagated into a solid point bounce back immediately and stay where they were. The flux through the membrane depended on the local pressure (advective contribution) and local concentration gradient (diffusive contribution) across the membrane. 

In the first set, diffusion theory calculations are performed if no thin regions, such as tank walls, are included. When such regions are considered, DTF calculations in the S approximation are done. In these calculations, the metal density is 18 g/cm, tite uranium isotopic composition is 93.19 wt% " U and 6.81 wt% and all regions are concentric spheres. The princlpu variables are the metal mass, the solution volume, and the concentration of uranium in solution. 

At penetration of monomer-diffuser to the gel-polymer matrix along the radiuses of spherical (or cylindrical) surface the front of spreading of gradient-former is compressed, homogenous gradient layer is formed with parallel stratified isorefraction (in the view of concentrated sphere or coaxial tubes) (Figure 2). 

First, the potential exhibits a maximum or a minimum at a point or axis of symmetry. These locations can be the centerline of a slab, the axis of a cylinder, or the center of a sphere. Figure 1.2a and Figure 1.2b consider two such cases. Figure 1.2a represents a spherical catalyst pellet in which a reactant of external concentration Q diffuses into the sphere and undergoes a reaction. Its concentration diminishes and attains a minimum at the center. Figure 1.2b considers laminar flow in a cylindrical pipe. Here the state variable in question is the axial velocity v, which rises from a value of zero at the wall to a maximum at the centerline before dropping back to zero at the other end of the diameter. Here, again, symmetry considerations dictate that this maximum must be located at the centerline of the conduit. 

For clarity, the remaining literature in this section is discussed in chronological order. Work by Phillies and Quinlan had been preceded by extensive optical probe diffusion studies of HPC water solutions Brown and Rymden used QELSS to examine 72 nm radius PSL spheres diffusing in solutions of hydroxyethylcellulose (HEC), hydroxypropylcellulose (HPC), carboxymethylcellulose (CMC), and polyacrylic acid(PAA)(56). The focus was polymer-induced cluster formation, indicated by the substantial decreases in Dp and increases in the second spectral cumulant as seen at very low (0.001 g/g) concentrations of HEC and HPC. These changes were substantially reversed by the addition of 0.15% Triton X-100. The Dp of spheres was reduced by the addition of small amounts of fully-charged pH 9 CMC, but addition of TX-lOO had no effect in CMC solutions. Brown and Rymden also examined sphere diffusion in nondilute polymer solutions. Relatively complex dependences of Dp on concentration were suppressed by the addition of TX-IOO. In the presence of TX-lOO, simple stretched-exponential concentration dependences were observed, but the second spectral cumulant still increased with increasing polymer concentration. 

Finally, theoretical calculations of the concentration dependence of tracer and mutual diffusion coefficients of nearly-dilute random coils and hard spheres, all based on the assumed dominance of hydrodynamic interactions, give reasonably quantitative agreement with experiment, suggesting that we correctly understand the forces driving sphere diffusion and the forces driving random-coil diffusion at modest concentrations, and that these forces are the same. 

In most electrochemical studies, one employs solutions where the concentration of the electroactive species, i, is 1 mM. With these concentrations, the diffusion flux of electroactive species to the electrode, /, is of the order of otjC, where OTj is the mass transfer coefficient (cm/s) and C is the bulk concentration (mol/cm ). With m, 10 cm/s, this produces fluxes of the order of 10 mol/s/ cm or 6 X10 " molecules/s/cm, producing currents of 10 " A/cm. Under these conditions, even with very small electrodes, one measures the behavior of large ensembles of molecules. However, if the concentration of electroactive species is dropped to 1 pM, these fluxes drop to/j=10 mol/s/ cm or 6 X10 molecules/s/cm with a current density of 10 A/cm. Thus, with an ultramicroelectrode (UME) with about a 10 pm size or area of about 10 cm, the number of molecules arriving by diffusion to the electrode is about 1/s. In our previous work, we showed that by using very small ( pm) nanoelectrocatalytic C or Au electrodes with relatively small background currents, that nanometer-size electrocatalytic NPs, for example, of Pt, amplify the current of an appropriate inner-sphere (IS) reaction (e.g., hydrazine oxidation or proton reduction) to the pA level, and the frequency, size, and shape of collision events could be investigated. More recent work with different approaches has shown that interactions of the NPs with the electrode can be detected, even for outer-sphere (OS) reactions, as described in later sections of this chapter. 

Consider two concentric spheres of radii and (Figure 5.12). A species diffuses through the annulus hmited by these two spheres under the action of a gradient of concentration imposed by the concentrations C, and at the two spherical surfaces these concentrations remain constant. We apply the steady state condition to the law of Pick taken in form [5.19], which leads to... 

This is not immediately applicable to electrophoresis, however. The solid particle with its fixed double layer (net charge Q) is moving relative to a solution in which the diffuse double layer is distributed (see electrical double layer). The latter is equivalent to a charge -Q spread out on a concentric sphere of radius where this is the thickness of the ionic atmosphere. The presence of this atmosphere reduces the mobility, and the potential at the surface of the particle, by the factor 1/(1+ Kr) so that in place of equation (E.16) the zeta-potential (see electrokinetic effects) is given by... 

Consider spherical molecules A and B having radii and Tb and diffusion coefficients Da and Db- First, suppose that B is fixed and that the rate of reaction is limited by the rate at which A molecules diffuse to the B molecule. We calculate the flux 7(A- B) of A molecules to one B molecule. Let a and b be the concentrations (in molecules/cm ) of A and B in the bulk, and let r be the radius of a sphere centered at the B molecule. The surface area of this sphere is Aitr, so by Pick s first law we obtain... 

Hulbert [77] discusses the consequences of the relatively large concentrations of lattice imperfections, including, perhaps, metastable phases and structural deformations, which may be present at the commencement of reaction but later diminish in concentration and importance. If it is assumed [475] that the rate of defect removal is inversely proportional to time (the Tammann treatment) and this effect is incorporated in the Valensi [470]—Carter [474] approach it is found that eqn. (12) is modified by replacement of t by In t. This equation is obeyed [77] by many spinel formation reactions. Zuravlev et al. [476] introduced the postulate that the rate of interface advance under diffusion control was also proportional to the amount of unreacted substance present and, assuming a contracting sphere (radius r) model... 

This equation reflects the rate of change with time of the concentration between parallel planes at points x and (x + dx) (which is equal to the difference in flux at the two planes). Fick s second law is vahd for the conditions assmned, namely planes parallel to one another and perpendicular to the direction of diffusion, i.e., conditions of linear diffusion. In contrast, for the case of diffusion toward a spherical electrode (where the lines of flux are not parallel but are perpendicular to segments of the sphere), Fick s second law has the form... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

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